# Definition:Strictly Well-Founded Relation

## Definition

Let $\struct {S, \RR}$ be a relational structure.

### Definition 1

$\RR$ is a **strictly well-founded relation on $S$** if and only if every non-empty subset of $S$ has a strictly minimal element under $\RR$.

### Definition 2

$\RR$ is a **strictly well-founded relation on $S$** if and only if:

- $\forall T: \paren {T \subseteq S \land T \ne \O} \implies \exists y \in T: \forall z \in T: \neg \paren {z \mathrel \RR y}$

where $\O$ is the empty set.

### Definition 3

Let $\RR$ be a **well-founded relation** which is also **antireflexive**.

Then $\RR$ is a **strictly well-founded relation on $S$**.

## Also known as

A **strictly well-founded relation** is also known in the literature as a **foundational relation**.

It is commonplace in the literature and on the internet to use the term **well-founded relation** for **strictly well-founded relation**.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more cumbersome and arguably more precise **strictly well-founded relation** in preference to all others.

Some sources do not hyphenate, and present the name as **strictly wellfounded relation**.

## Also see

- Results about
**well-founded relations**can be found here.